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I'm told that when a magnetic dipole is placed in a magnetic field $\vec{B}$, it experiences a torque $\vec{\tau} = \vec\mu \times \vec{B}$ and an associated energy $ H = -\vec{\mu} \cdot \vec{B} $, where $\vec{\mu}$ is the magnetic dipole moment, pointing in the same direction as how the dipole is oriented in space.

I thought at first the associated energy might be the total amount of work you'd have to do to spin the dipole from equilibrium to whatever orientation you care about, but this doesn't agree with the math. If $\vec{\mu}$ points in the same direction as $\vec{B}$, you get a non zero value. What on earth does this associated energy actually mean?

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You're mostly correct, except for the expectation that the minimum energy orientation is used for zero potential energy. Instead, the zero point is when the spin and field are perpendicular.

This suggests (correctly) that it will take energy to remove a dipole from a magnetic field after they have become parallel, as the potential energy in a zero-strength magnetic field is greater than the potential energy when it is aligned in a magnetic field.

This should be no odder than using the general gravitational potential energy equation, where the potential energy of a system of masses is negative.

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